Question

Portfolio returns. The Capital Asset Pricing Model is a financial model that assumes returns on a portfolio are normally distributed. Suppose a portfolio has an average annual return of 17.4% (i.e. an average gain of 17.4%) with a standard deviation of 39%. A return of 0% means the value of the portfolio doesn't change, a negative return means that the portfolio loses money, and a positive return means that the portfolio gains money. Round all answers to 4 decimal places.

a. What percent of years does this portfolio lose money, i.e. have a return less than 0%?  %

b. What is the cutoff for the highest 17% of annual returns with this portfolio?  %

Answer 1

The percent of years does the portfolio lose money. That is, find the probability P(X<0)

Let X be the random variable defined by returns on a portfolio follows normal distribution with mean(μ) 17.4 % and standard deviation(σ) 39%.

The probability P(X<0) is,

P(X<0)=P(x-17.4/39<0-17.4/39)

=P(z<-0.4461)

=0.3277 ============>> Use ''=NORMSDIST(-0.4461)'' in Excl.

The percent of the year’s portfolio lose money, that is have a return less than 0% is 32.77%

b.

The cutoff for the highest 17% of annual returns with this portfolio is obtained below:

P(X>x)=0.17

P(X<x)=1-0.17

=0.83

From the “standard Normal table”, the area covered for value of 0.83 is obtained at . z=0.9541

The cutoff for the highest 17% of annual returns with this portfolio is,

Z=(x-mean)/standard deviation.

0.9541=(x-17.4)/39

x=17.4+0.9541*39

=54.61

The cutoff for the highest 17% of annual returns with this portfolio is 54.61%